Let $G$ be a group scheme locally finite type and smooth over a base scheme $S$, and assume $S$ is normal and integral.
Then does the set of (geometrical) connected components of a group scheme form a group? or even point of a group scheme over $S$?
If $S=Spec k $ where $k$ is a field, then this is true and we have a theory of $\pi_0(G)$ using etale algebras over a field. I wonder what will happen for the generic case.